PHY3707
Assignment 5
Unique number: 657543
Due 18 August 2025
, PHY3707 Assignment 5 — Due Date: 18 August 2025
PHY3707 Assignment 5
Due Date: 18 August 2025
6.9 [10 marks]
Problem: Show that the fraction of electrons within kB T of the Fermi
3kB T
level is equal to 2εF , if the density of states satisfies D(ε) ∼ ε1/2 .
Solution:
Step 1: Density of states and fraction of electrons
Given the density of states
D(ε) = Cε1/2 ,
where C is a constant. The total number of electrons up to the Fermi
level εF is proportional to
Z εF
2 3/2
N∝ Cε1/2 dε = C · εF .
0 3
The number of electrons within kB T of the Fermi level is
Z εF +kB T
∆N ∝ Cε1/2 dε.
εF −kB T
Step 2: Approximating the integral
Since kB T ≪ εF , approximate the integral using the difference:
2 3/2 2 3/2
∆N ≈ C (εF + kB T ) − (εF − kB T ) .
3 3
Page 1
Assignment 5
Unique number: 657543
Due 18 August 2025
, PHY3707 Assignment 5 — Due Date: 18 August 2025
PHY3707 Assignment 5
Due Date: 18 August 2025
6.9 [10 marks]
Problem: Show that the fraction of electrons within kB T of the Fermi
3kB T
level is equal to 2εF , if the density of states satisfies D(ε) ∼ ε1/2 .
Solution:
Step 1: Density of states and fraction of electrons
Given the density of states
D(ε) = Cε1/2 ,
where C is a constant. The total number of electrons up to the Fermi
level εF is proportional to
Z εF
2 3/2
N∝ Cε1/2 dε = C · εF .
0 3
The number of electrons within kB T of the Fermi level is
Z εF +kB T
∆N ∝ Cε1/2 dε.
εF −kB T
Step 2: Approximating the integral
Since kB T ≪ εF , approximate the integral using the difference:
2 3/2 2 3/2
∆N ≈ C (εF + kB T ) − (εF − kB T ) .
3 3
Page 1
